Use the discriminant to tell whether the solutions of this equation are real or complex. \[ x^{2}-4 x+1=0 \]

To determine whether the solutions of the equation \( x^{2}-4 x+1=0 \) are real or complex, we will calculate the discriminant using the formula \( D = b^2 - 4ac \). Here, \( a = 1 \), \( b = -4 \), and \( c = 1 \). Calculating the discriminant: \[ D = (-4)^2 - 4(1)(1) = 16 - 4 = 12 \] Since the discriminant \( D = 12 \) is positive, the solutions to the equation are real and distinct. Real-world applications of discriminants are fascinating! For example, engineers use them in designing parabolas for bridges or archways to ensure the structure can bear loads. If the discriminant indicates complex solutions, it might mean an impractical design, guiding engineers to rethink their approach. A common mistake when using the discriminant is forgetting to square the \( b \) value, especially if the quadratic is complex! Another error is miscalculating \( 4ac \) or overlooking the signs in the formula. Always double-check your computations, and you’ll hit the jackpot of solutions!